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Geometry 1 Unit 6

Geometry 1 Unit 6. Quadrilaterals. Geometry 1 Unit 6. 6.1 Polygons. Polygons. Polygon A closed figure in a plane Formed by connecting line segments endpoint to endpoint Each segment intersects exactly 2 others Classified by the number of sides they have

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Geometry 1 Unit 6

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  1. Geometry 1 Unit 6 Quadrilaterals

  2. Geometry 1 Unit 6 6.1 Polygons

  3. Polygons • Polygon • A closed figure in a plane • Formed by connecting line segments endpoint to endpoint • Each segment intersects exactly 2 others • Classified by the number of sides they have • Named by listing vertices in consecutive order • Sides • Line segments in a polygon • Vertex • Each endpoint in a polygon

  4. Polygons polygons not polygons

  5. A E B C D Polygons Pentagon ABCDE or pentagon CDEAB

  6. Polygons Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon n n-gon (a 19 sided polygon is a 19-gon)

  7. Polygons • Diagonals • Line segments that connect non-consecutive vertices.

  8. Polygons • Convex polygons • Polygons with no diagonals on the outside of the polygon

  9. Polygons • Concave polygons • A polygon is concave if at least one diagonal is outside the polygon • These are also called nonconvex.

  10. Polygons • Example 1 • Identify the polygon and state whether it is convex or concave.

  11. Polygons • Equilateral Polygon • all sides the same length • Equiangular Polygon • all angles equal measure • Regular Polygon • equilateral and equiangular

  12. Polygons Equilateral Equiangular Regular

  13. Polygons • Example 2 • Decide whether the polygon is regular.

  14. 2 1 4 3 Polygons • Interior Angles of a Quadrilateral Theorem • The sum of the measures of the interior angles of a quadrilateral is 360°. m1 + m2 + m3 + m4 = 360°

  15. H G x 55° x E F Polygons • Example 3 • Find mF, mG, and mH.

  16. 100° 120° 3x – 5 2x + 30 Polygons • Example 4 • Use the information in the diagram to solve for x

  17. Geometry 1 Unit 6 6.2 Properties of Parallelograms

  18. Properties of Parallelograms • Parallelogram • Quadrilateral with two pairs of parallel sides.

  19. Q R PQ  RS and SP  QR P S Properties of Parallelograms • Opposite Sides of a Parallelogram Theorem • If a quadrilateral is a parallelogram, then its opposite sides are congruent.

  20. Q R P S Properties of Parallelograms • Opposite Angles in a Parallelogram Theorem • If a quadrilateral is a parallelogram, then its opposite angles are congruent. P  R and Q  S

  21. Add to equal 180° Properties of Parallelograms • Consecutive Angles in a Parallelogram Theorem • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary R Q mP + mQ = 180° mQ + mR = 180° mR + mS = 180° mS + mP = 180° S P

  22. Q R QM  SM and PM  RM M P S Properties of Parallelograms • Diagonals in a Parallelogram Theorem • If a quadrilateral is a parallelogram, then its diagonals bisect each other.

  23. K J  6 8   L G  H Properties of Parallelograms • Example 1 • GHJK is a parallelogram. Find each unknown length • JH • LH

  24. Properties of Parallelograms • Example 2 • In ABCD, mC = 105°. Find the measure of each angle. • mA • mD

  25. Properties of Parallelograms • Example 3 • WXYZ is a parallelogram. Find the value of x. Z Y  3x + 18°   4x – 9°  W X

  26. A B 1 2 4 3 D C Properties of Parallelograms • Example 4 • Given: • ABCD is a parallelogram. • Prove: • 2  4

  27. A B C F E D Properties of Parallelograms • Example 5 • Given: • ACDF is a parallelogram. • ABDE is a parallelogram. • Prove: • ∆BCD  ∆EFA

  28. Properties of Parallelograms • Example 6 • A four-sided concrete slab has consecutive angle measures of 85°, 94°, 85°, and 96°. Is the slab a parallelogram? Explain.

  29. Geometry 1 Unit 6 6.3 Proving Quadrilaterals are Parallelograms

  30. Proving Quadrilaterals are Parallelograms • Investigating Properties of Parallelograms • Cut 4 straws to form two congruent pairs. • Partly unbend two paperclips, link their smaller ends, and insert the larger ends into two cut straws. Join the rest of the straws to form a quadrilateral with opposite sides congruent. • Change the angles of your quadrilateral. Is your quadrilateral a parallelogram?

  31. A B D C Proving Quadrilaterals are Parallelograms • Converse of the Opposite Sides of a Parallelogram Theorem • If a opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram

  32. A B D C Proving Quadrilaterals are Parallelograms • Converse of the Opposite Angles in a Parallelogram Theorem • If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

  33. B A x° (180 – x)° x° C D Proving Quadrilaterals are Parallelograms • Converse of the Consecutive Angles in a Parallelogram Theorem • If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. ABCD is a parallelogram

  34. A B M D C Proving Quadrilaterals are Parallelograms • Converse of the Diagonals in a Parallelogram Theorem • If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. ABCD is a parallelogram

  35. P Q T S R Proving Quadrilaterals are Parallelograms • Example 1 • Given: • ∆PQT  ∆RST • Prove: • PQRS is a parallelogram.

  36. Proving Quadrilaterals are Parallelograms • Example 2 • A gate is braced as shown. How do you know that opposite sides of the gate are congruent?

  37. B C   A D Proving Quadrilaterals are Parallelograms • Congruent and Parallel Sides Theorem • If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

  38. Proving Quadrilaterals are Parallelograms • To determine if a quadrilateral is a parallelogram, you need to know one of the following: • Opposite sides are parallel • Opposite sides are congruent • Opposite angles are congruent • An angle is supplementary with both of its consecutive angles • Diagonals bisect each other • One pair of sides is both parallel and congruent

  39. Proving Quadrilaterals are Parallelograms • Example 3 • Show that A(-1,2), B(3,2), C(1,-2), and D(-3,-2) are the vertices of a parallelogram.

  40. Geometry 1 Unit 6 6.4 Rhombuses, Rectangles, and Squares

  41. Rhombuses, Rectangles, and Squares • Rectangle • Parallelogram with four congruent angles • Rhombus • Parallelogram with four congruent sides • Square • Parallelogram with four congruent angles and four congruent sides

  42. Rhombuses, Rectangles, and Squares • Example 1 • Decide if each statement is always, sometimes or never true. • A rhombus is a rectangle • A parallelogram is a rectangle • A rectangle is a square • A square is a rhombus

  43. F R O G Rhombuses, Rectangles, and Squares • Example 2 • Given FROG is a rectangle, what else do you know about FROG?

  44. Rhombuses, Rectangles, and Squares • Example 3 • EFGH is a rectangle. K is the midpoint of FH. EG = 8z – 16, • What is the measure of segment EK? • What is the measure of segment GK?

  45. Rhombuses, Rectangles, and Squares • Rhombus Corollary • A quadrilateral is a rhombus if and only if it has four congruent sides.

  46. Rhombuses, Rectangles, and Squares • Rectangle Corollary • A quadrilateral is a rectangle if and only if it has four right angles.

  47. Rhombuses, Rectangles, and Squares • Square Corollary • A quadrilateral is a square if and only if it is a rhombus and a rectangle.

  48. B C  ABCD is a rhombus if and only if AC BD.    A D Rhombuses, Rectangles, and Squares • Perpendicular Diagonals of a Rhombus Theorem • A parallelogram is a rhombus if and only if its diagonals are perpendicular.

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